LMFDB - Embedded newform 6008.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9741215344$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.22705 q^{3} +2.92731 q^{5} +0.525757 q^{7} +7.41384 q^{9} +O(q^{10})$	$q-3.22705 q^{3} +2.92731 q^{5} +0.525757 q^{7} +7.41384 q^{9} -1.19339 q^{11} -1.45885 q^{13} -9.44658 q^{15} +4.32149 q^{17} +2.66853 q^{19} -1.69664 q^{21} -0.287634 q^{23} +3.56917 q^{25} -14.2437 q^{27} +4.93433 q^{29} -5.71935 q^{31} +3.85112 q^{33} +1.53906 q^{35} -3.78706 q^{37} +4.70777 q^{39} +10.2624 q^{41} +3.99765 q^{43} +21.7026 q^{45} +1.23518 q^{47} -6.72358 q^{49} -13.9457 q^{51} +12.0361 q^{53} -3.49342 q^{55} -8.61147 q^{57} +12.3670 q^{59} -3.71482 q^{61} +3.89788 q^{63} -4.27050 q^{65} +2.87365 q^{67} +0.928209 q^{69} -4.40772 q^{71} +1.18677 q^{73} -11.5179 q^{75} -0.627433 q^{77} -8.82191 q^{79} +23.7235 q^{81} +14.4423 q^{83} +12.6504 q^{85} -15.9233 q^{87} -7.97365 q^{89} -0.767000 q^{91} +18.4566 q^{93} +7.81162 q^{95} -10.5153 q^{97} -8.84759 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.22705	−1.86314	−0.931568	−	0.363566i	$-0.881559\pi$
$3$	−3.22705	−1.86314	−0.931568	+	0.363566i	$0.881559\pi$
$4$	0	0
$5$	2.92731	1.30913	0.654567	−	0.756004i	$-0.272851\pi$
$5$	2.92731	1.30913	0.654567	+	0.756004i	$0.272851\pi$
$6$	0	0
$7$	0.525757	0.198718	0.0993588	−	0.995052i	$-0.468321\pi$
$7$	0.525757	0.198718	0.0993588	+	0.995052i	$0.468321\pi$
$8$	0	0
$9$	7.41384	2.47128
$10$	0	0
$11$	−1.19339	−0.359820	−0.179910	−	0.983683i	$-0.557581\pi$
$11$	−1.19339	−0.359820	−0.179910	+	0.983683i	$0.557581\pi$
$12$	0	0
$13$	−1.45885	−0.404611	−0.202306	−	0.979322i	$-0.564843\pi$
$13$	−1.45885	−0.404611	−0.202306	+	0.979322i	$0.564843\pi$
$14$	0	0
$15$	−9.44658	−2.43910
$16$	0	0
$17$	4.32149	1.04812	0.524058	−	0.851683i	$-0.324418\pi$
$17$	4.32149	1.04812	0.524058	+	0.851683i	$0.324418\pi$
$18$	0	0
$19$	2.66853	0.612202	0.306101	−	0.951999i	$-0.400975\pi$
$19$	2.66853	0.612202	0.306101	+	0.951999i	$0.400975\pi$
$20$	0	0
$21$	−1.69664	−0.370238
$22$	0	0
$23$	−0.287634	−0.0599758	−0.0299879	−	0.999550i	$-0.509547\pi$
$23$	−0.287634	−0.0599758	−0.0299879	+	0.999550i	$0.509547\pi$
$24$	0	0
$25$	3.56917	0.713833
$26$	0	0
$27$	−14.2437	−2.74119
$28$	0	0
$29$	4.93433	0.916282	0.458141	−	0.888880i	$-0.348516\pi$
$29$	4.93433	0.916282	0.458141	+	0.888880i	$0.348516\pi$
$30$	0	0
$31$	−5.71935	−1.02723	−0.513613	−	0.858022i	$-0.671693\pi$
$31$	−5.71935	−1.02723	−0.513613	+	0.858022i	$0.671693\pi$
$32$	0	0
$33$	3.85112	0.670394
$34$	0	0
$35$	1.53906	0.260148
$36$	0	0
$37$	−3.78706	−0.622588	−0.311294	−	0.950314i	$-0.600762\pi$
$37$	−3.78706	−0.622588	−0.311294	+	0.950314i	$0.600762\pi$
$38$	0	0
$39$	4.70777	0.753846
$40$	0	0
$41$	10.2624	1.60271	0.801355	−	0.598189i	$-0.204113\pi$
$41$	10.2624	1.60271	0.801355	+	0.598189i	$0.204113\pi$
$42$	0	0
$43$	3.99765	0.609636	0.304818	−	0.952411i	$-0.401404\pi$
$43$	3.99765	0.609636	0.304818	+	0.952411i	$0.401404\pi$
$44$	0	0
$45$	21.7026	3.23524
$46$	0	0
$47$	1.23518	0.180169	0.0900846	−	0.995934i	$-0.471286\pi$
$47$	1.23518	0.180169	0.0900846	+	0.995934i	$0.471286\pi$
$48$	0	0
$49$	−6.72358	−0.960511
$50$	0	0
$51$	−13.9457	−1.95278
$52$	0	0
$53$	12.0361	1.65329	0.826643	−	0.562727i	$-0.190248\pi$
$53$	12.0361	1.65329	0.826643	+	0.562727i	$0.190248\pi$
$54$	0	0
$55$	−3.49342	−0.471053
$56$	0	0
$57$	−8.61147	−1.14062
$58$	0	0
$59$	12.3670	1.61004	0.805022	−	0.593245i	$-0.202154\pi$
$59$	12.3670	1.61004	0.805022	+	0.593245i	$0.202154\pi$
$60$	0	0
$61$	−3.71482	−0.475634	−0.237817	−	0.971310i	$-0.576432\pi$
$61$	−3.71482	−0.475634	−0.237817	+	0.971310i	$0.576432\pi$
$62$	0	0
$63$	3.89788	0.491087
$64$	0	0
$65$	−4.27050	−0.529691
$66$	0	0
$67$	2.87365	0.351072	0.175536	−	0.984473i	$-0.443834\pi$
$67$	2.87365	0.351072	0.175536	+	0.984473i	$0.443834\pi$
$68$	0	0
$69$	0.928209	0.111743
$70$	0	0
$71$	−4.40772	−0.523100	−0.261550	−	0.965190i	$-0.584234\pi$
$71$	−4.40772	−0.523100	−0.261550	+	0.965190i	$0.584234\pi$
$72$	0	0
$73$	1.18677	0.138901	0.0694504	−	0.997585i	$-0.477875\pi$
$73$	1.18677	0.138901	0.0694504	+	0.997585i	$0.477875\pi$
$74$	0	0
$75$	−11.5179	−1.32997
$76$	0	0
$77$	−0.627433	−0.0715026
$78$	0	0
$79$	−8.82191	−0.992542	−0.496271	−	0.868168i	$-0.665298\pi$
$79$	−8.82191	−0.992542	−0.496271	+	0.868168i	$0.665298\pi$
$80$	0	0
$81$	23.7235	2.63594
$82$	0	0
$83$	14.4423	1.58525	0.792625	−	0.609709i	$-0.208714\pi$
$83$	14.4423	1.58525	0.792625	+	0.609709i	$0.208714\pi$
$84$	0	0
$85$	12.6504	1.37212
$86$	0	0
$87$	−15.9233	−1.70716
$88$	0	0
$89$	−7.97365	−0.845205	−0.422603	−	0.906315i	$-0.638883\pi$
$89$	−7.97365	−0.845205	−0.422603	+	0.906315i	$0.638883\pi$
$90$	0	0
$91$	−0.767000	−0.0804034
$92$	0	0
$93$	18.4566	1.91386
$94$	0	0
$95$	7.81162	0.801455
$96$	0	0
$97$	−10.5153	−1.06767	−0.533835	−	0.845588i	$-0.679250\pi$
$97$	−10.5153	−1.06767	−0.533835	+	0.845588i	$0.679250\pi$
$98$	0	0
$99$	−8.84759	−0.889216
$100$	0	0
$101$	−8.56150	−0.851901	−0.425950	−	0.904747i	$-0.640060\pi$
$101$	−8.56150	−0.851901	−0.425950	+	0.904747i	$0.640060\pi$
$102$	0	0
$103$	10.0164	0.986950	0.493475	−	0.869760i	$-0.335726\pi$
$103$	10.0164	0.986950	0.493475	+	0.869760i	$0.335726\pi$
$104$	0	0
$105$	−4.96661	−0.484691
$106$	0	0
$107$	13.4499	1.30025	0.650125	−	0.759827i	$-0.274716\pi$
$107$	13.4499	1.30025	0.650125	+	0.759827i	$0.274716\pi$
$108$	0	0
$109$	18.3512	1.75773	0.878865	−	0.477070i	$-0.158301\pi$
$109$	18.3512	1.75773	0.878865	+	0.477070i	$0.158301\pi$
$110$	0	0
$111$	12.2210	1.15997
$112$	0	0
$113$	−4.46354	−0.419895	−0.209947	−	0.977713i	$-0.567329\pi$
$113$	−4.46354	−0.419895	−0.209947	+	0.977713i	$0.567329\pi$
$114$	0	0
$115$	−0.841995	−0.0785164
$116$	0	0
$117$	−10.8157	−0.999908
$118$	0	0
$119$	2.27206	0.208279
$120$	0	0
$121$	−9.57582	−0.870529
$122$	0	0
$123$	−33.1171	−2.98607
$124$	0	0
$125$	−4.18850	−0.374631
$126$	0	0
$127$	−10.9047	−0.967635	−0.483817	−	0.875169i	$-0.660750\pi$
$127$	−10.9047	−0.967635	−0.483817	+	0.875169i	$0.660750\pi$
$128$	0	0
$129$	−12.9006	−1.13583
$130$	0	0
$131$	−12.7298	−1.11221	−0.556106	−	0.831112i	$-0.687705\pi$
$131$	−12.7298	−1.11221	−0.556106	+	0.831112i	$0.687705\pi$
$132$	0	0
$133$	1.40300	0.121655
$134$	0	0
$135$	−41.6957	−3.58859
$136$	0	0
$137$	−9.35071	−0.798885	−0.399443	−	0.916758i	$-0.630796\pi$
$137$	−9.35071	−0.798885	−0.399443	+	0.916758i	$0.630796\pi$
$138$	0	0
$139$	−18.3516	−1.55656	−0.778281	−	0.627916i	$-0.783908\pi$
$139$	−18.3516	−1.55656	−0.778281	+	0.627916i	$0.783908\pi$
$140$	0	0
$141$	−3.98598	−0.335680
$142$	0	0
$143$	1.74097	0.145587
$144$	0	0
$145$	14.4443	1.19954
$146$	0	0
$147$	21.6973	1.78956
$148$	0	0
$149$	−3.53933	−0.289953	−0.144977	−	0.989435i	$-0.546311\pi$
$149$	−3.53933	−0.289953	−0.144977	+	0.989435i	$0.546311\pi$
$150$	0	0
$151$	2.32285	0.189031	0.0945153	−	0.995523i	$-0.469870\pi$
$151$	2.32285	0.189031	0.0945153	+	0.995523i	$0.469870\pi$
$152$	0	0
$153$	32.0388	2.59019
$154$	0	0
$155$	−16.7423	−1.34478
$156$	0	0
$157$	−2.35205	−0.187714	−0.0938569	−	0.995586i	$-0.529920\pi$
$157$	−2.35205	−0.187714	−0.0938569	+	0.995586i	$0.529920\pi$
$158$	0	0
$159$	−38.8411	−3.08030
$160$	0	0
$161$	−0.151226	−0.0119183
$162$	0	0
$163$	6.41777	0.502678	0.251339	−	0.967899i	$-0.419129\pi$
$163$	6.41777	0.502678	0.251339	+	0.967899i	$0.419129\pi$
$164$	0	0
$165$	11.2734	0.877637
$166$	0	0
$167$	22.5965	1.74857	0.874287	−	0.485410i	$-0.161330\pi$
$167$	22.5965	1.74857	0.874287	+	0.485410i	$0.161330\pi$
$168$	0	0
$169$	−10.8718	−0.836290
$170$	0	0
$171$	19.7840	1.51292
$172$	0	0
$173$	10.4379	0.793577	0.396788	−	0.917910i	$-0.370125\pi$
$173$	10.4379	0.793577	0.396788	+	0.917910i	$0.370125\pi$
$174$	0	0
$175$	1.87652	0.141851
$176$	0	0
$177$	−39.9088	−2.99973
$178$	0	0
$179$	7.49285	0.560042	0.280021	−	0.959994i	$-0.409659\pi$
$179$	7.49285	0.560042	0.280021	+	0.959994i	$0.409659\pi$
$180$	0	0
$181$	18.7890	1.39657	0.698287	−	0.715818i	$-0.253946\pi$
$181$	18.7890	1.39657	0.698287	+	0.715818i	$0.253946\pi$
$182$	0	0
$183$	11.9879	0.886171
$184$	0	0
$185$	−11.0859	−0.815052
$186$	0	0
$187$	−5.15722	−0.377133
$188$	0	0
$189$	−7.48871	−0.544724
$190$	0	0
$191$	−10.1590	−0.735080	−0.367540	−	0.930008i	$-0.619800\pi$
$191$	−10.1590	−0.735080	−0.367540	+	0.930008i	$0.619800\pi$
$192$	0	0
$193$	8.59068	0.618370	0.309185	−	0.951002i	$-0.399944\pi$
$193$	8.59068	0.618370	0.309185	+	0.951002i	$0.399944\pi$
$194$	0	0
$195$	13.7811	0.986886
$196$	0	0
$197$	17.9621	1.27975	0.639875	−	0.768479i	$-0.278986\pi$
$197$	17.9621	1.27975	0.639875	+	0.768479i	$0.278986\pi$
$198$	0	0
$199$	−4.24759	−0.301104	−0.150552	−	0.988602i	$-0.548105\pi$
$199$	−4.24759	−0.301104	−0.150552	+	0.988602i	$0.548105\pi$
$200$	0	0
$201$	−9.27341	−0.654096
$202$	0	0
$203$	2.59426	0.182081
$204$	0	0
$205$	30.0411	2.09816
$206$	0	0
$207$	−2.13247	−0.148217
$208$	0	0
$209$	−3.18459	−0.220283
$210$	0	0
$211$	−15.0398	−1.03538	−0.517690	−	0.855568i	$-0.673208\pi$
$211$	−15.0398	−1.03538	−0.517690	+	0.855568i	$0.673208\pi$
$212$	0	0
$213$	14.2239	0.974608
$214$	0	0
$215$	11.7024	0.798095
$216$	0	0
$217$	−3.00699	−0.204128
$218$	0	0
$219$	−3.82976	−0.258791
$220$	0	0
$221$	−6.30439	−0.424079
$222$	0	0
$223$	−10.0775	−0.674842	−0.337421	−	0.941354i	$-0.609555\pi$
$223$	−10.0775	−0.674842	−0.337421	+	0.941354i	$0.609555\pi$
$224$	0	0
$225$	26.4612	1.76408
$226$	0	0
$227$	−12.0629	−0.800642	−0.400321	−	0.916375i	$-0.631101\pi$
$227$	−12.0629	−0.800642	−0.400321	+	0.916375i	$0.631101\pi$
$228$	0	0
$229$	−9.47307	−0.625998	−0.312999	−	0.949753i	$-0.601334\pi$
$229$	−9.47307	−0.625998	−0.312999	+	0.949753i	$0.601334\pi$
$230$	0	0
$231$	2.02476	0.133219
$232$	0	0
$233$	1.99752	0.130862	0.0654308	−	0.997857i	$-0.479158\pi$
$233$	1.99752	0.130862	0.0654308	+	0.997857i	$0.479158\pi$
$234$	0	0
$235$	3.61575	0.235866
$236$	0	0
$237$	28.4687	1.84924
$238$	0	0
$239$	14.2781	0.923572	0.461786	−	0.886991i	$-0.347209\pi$
$239$	14.2781	0.923572	0.461786	+	0.886991i	$0.347209\pi$
$240$	0	0
$241$	12.4854	0.804254	0.402127	−	0.915584i	$-0.368271\pi$
$241$	12.4854	0.804254	0.402127	+	0.915584i	$0.368271\pi$
$242$	0	0
$243$	−33.8258	−2.16993
$244$	0	0
$245$	−19.6820	−1.25744
$246$	0	0
$247$	−3.89297	−0.247704
$248$	0	0
$249$	−46.6061	−2.95354
$250$	0	0
$251$	−20.1642	−1.27275	−0.636376	−	0.771379i	$-0.719567\pi$
$251$	−20.1642	−1.27275	−0.636376	+	0.771379i	$0.719567\pi$
$252$	0	0
$253$	0.343259	0.0215805
$254$	0	0
$255$	−40.8233	−2.55646
$256$	0	0
$257$	−6.96476	−0.434450	−0.217225	−	0.976122i	$-0.569700\pi$
$257$	−6.96476	−0.434450	−0.217225	+	0.976122i	$0.569700\pi$
$258$	0	0
$259$	−1.99107	−0.123719
$260$	0	0
$261$	36.5823	2.26439
$262$	0	0
$263$	16.4273	1.01295	0.506475	−	0.862255i	$-0.330948\pi$
$263$	16.4273	1.01295	0.506475	+	0.862255i	$0.330948\pi$
$264$	0	0
$265$	35.2334	2.16437
$266$	0	0
$267$	25.7313	1.57473
$268$	0	0
$269$	11.1581	0.680323	0.340162	−	0.940367i	$-0.389518\pi$
$269$	11.1581	0.680323	0.340162	+	0.940367i	$0.389518\pi$
$270$	0	0
$271$	27.0322	1.64209	0.821046	−	0.570862i	$-0.193391\pi$
$271$	27.0322	1.64209	0.821046	+	0.570862i	$0.193391\pi$
$272$	0	0
$273$	2.47514	0.149803
$274$	0	0
$275$	−4.25940	−0.256852
$276$	0	0
$277$	10.2355	0.614990	0.307495	−	0.951550i	$-0.400509\pi$
$277$	10.2355	0.614990	0.307495	+	0.951550i	$0.400509\pi$
$278$	0	0
$279$	−42.4023	−2.53856
$280$	0	0
$281$	6.72767	0.401339	0.200670	−	0.979659i	$-0.435688\pi$
$281$	6.72767	0.401339	0.200670	+	0.979659i	$0.435688\pi$
$282$	0	0
$283$	6.08609	0.361780	0.180890	−	0.983503i	$-0.442102\pi$
$283$	6.08609	0.361780	0.180890	+	0.983503i	$0.442102\pi$
$284$	0	0
$285$	−25.2085	−1.49322
$286$	0	0
$287$	5.39551	0.318487
$288$	0	0
$289$	1.67528	0.0985458
$290$	0	0
$291$	33.9335	1.98922
$292$	0	0
$293$	−13.4113	−0.783498	−0.391749	−	0.920072i	$-0.628130\pi$
$293$	−13.4113	−0.783498	−0.391749	+	0.920072i	$0.628130\pi$
$294$	0	0
$295$	36.2020	2.10776
$296$	0	0
$297$	16.9982	0.986337
$298$	0	0
$299$	0.419614	0.0242669
$300$	0	0
$301$	2.10179	0.121145
$302$	0	0
$303$	27.6284	1.58721
$304$	0	0
$305$	−10.8744	−0.622669
$306$	0	0
$307$	29.6956	1.69482	0.847408	−	0.530943i	$-0.178162\pi$
$307$	29.6956	1.69482	0.847408	+	0.530943i	$0.178162\pi$
$308$	0	0
$309$	−32.3236	−1.83882
$310$	0	0
$311$	−2.85559	−0.161926	−0.0809629	−	0.996717i	$-0.525800\pi$
$311$	−2.85559	−0.161926	−0.0809629	+	0.996717i	$0.525800\pi$
$312$	0	0
$313$	6.24002	0.352707	0.176353	−	0.984327i	$-0.443570\pi$
$313$	6.24002	0.352707	0.176353	+	0.984327i	$0.443570\pi$
$314$	0	0
$315$	11.4103	0.642899
$316$	0	0
$317$	2.52438	0.141783	0.0708915	−	0.997484i	$-0.477416\pi$
$317$	2.52438	0.141783	0.0708915	+	0.997484i	$0.477416\pi$
$318$	0	0
$319$	−5.88857	−0.329697
$320$	0	0
$321$	−43.4034	−2.42254
$322$	0	0
$323$	11.5320	0.641659
$324$	0	0
$325$	−5.20687	−0.288825
$326$	0	0
$327$	−59.2204	−3.27489
$328$	0	0
$329$	0.649404	0.0358028
$330$	0	0
$331$	−10.7439	−0.590538	−0.295269	−	0.955414i	$-0.595409\pi$
$331$	−10.7439	−0.590538	−0.295269	+	0.955414i	$0.595409\pi$
$332$	0	0
$333$	−28.0766	−1.53859
$334$	0	0
$335$	8.41208	0.459601
$336$	0	0
$337$	19.8614	1.08192	0.540959	−	0.841049i	$-0.318061\pi$
$337$	19.8614	1.08192	0.540959	+	0.841049i	$0.318061\pi$
$338$	0	0
$339$	14.4041	0.782321
$340$	0	0
$341$	6.82541	0.369616
$342$	0	0
$343$	−7.21527	−0.389588
$344$	0	0
$345$	2.71716	0.146287
$346$	0	0
$347$	18.4429	0.990068	0.495034	−	0.868874i	$-0.335156\pi$
$347$	18.4429	0.990068	0.495034	+	0.868874i	$0.335156\pi$
$348$	0	0
$349$	21.6946	1.16129	0.580643	−	0.814159i	$-0.302801\pi$
$349$	21.6946	1.16129	0.580643	+	0.814159i	$0.302801\pi$
$350$	0	0
$351$	20.7793	1.10912
$352$	0	0
$353$	1.41973	0.0755645	0.0377823	−	0.999286i	$-0.487971\pi$
$353$	1.41973	0.0755645	0.0377823	+	0.999286i	$0.487971\pi$
$354$	0	0
$355$	−12.9028	−0.684809
$356$	0	0
$357$	−7.33203	−0.388052
$358$	0	0
$359$	−0.356449	−0.0188127	−0.00940633	−	0.999956i	$-0.502994\pi$
$359$	−0.356449	−0.0188127	−0.00940633	+	0.999956i	$0.502994\pi$
$360$	0	0
$361$	−11.8790	−0.625208
$362$	0	0
$363$	30.9016	1.62192
$364$	0	0
$365$	3.47405	0.181840
$366$	0	0
$367$	2.71839	0.141899	0.0709495	−	0.997480i	$-0.477397\pi$
$367$	2.71839	0.141899	0.0709495	+	0.997480i	$0.477397\pi$
$368$	0	0
$369$	76.0834	3.96074
$370$	0	0
$371$	6.32807	0.328537
$372$	0	0
$373$	−30.0894	−1.55797	−0.778986	−	0.627041i	$-0.784266\pi$
$373$	−30.0894	−1.55797	−0.778986	+	0.627041i	$0.784266\pi$
$374$	0	0
$375$	13.5165	0.697988
$376$	0	0
$377$	−7.19843	−0.370738
$378$	0	0
$379$	−18.6583	−0.958412	−0.479206	−	0.877703i	$-0.659075\pi$
$379$	−18.6583	−0.958412	−0.479206	+	0.877703i	$0.659075\pi$
$380$	0	0
$381$	35.1900	1.80284
$382$	0	0
$383$	12.0271	0.614556	0.307278	−	0.951620i	$-0.400582\pi$
$383$	12.0271	0.614556	0.307278	+	0.951620i	$0.400582\pi$
$384$	0	0
$385$	−1.83669	−0.0936065
$386$	0	0
$387$	29.6379	1.50658
$388$	0	0
$389$	25.8206	1.30916	0.654578	−	0.755994i	$-0.272846\pi$
$389$	25.8206	1.30916	0.654578	+	0.755994i	$0.272846\pi$
$390$	0	0
$391$	−1.24301	−0.0628616
$392$	0	0
$393$	41.0798	2.07220
$394$	0	0
$395$	−25.8245	−1.29937
$396$	0	0
$397$	8.09867	0.406461	0.203230	−	0.979131i	$-0.434856\pi$
$397$	8.09867	0.406461	0.203230	+	0.979131i	$0.434856\pi$
$398$	0	0
$399$	−4.52754	−0.226661
$400$	0	0
$401$	7.64427	0.381736	0.190868	−	0.981616i	$-0.438870\pi$
$401$	7.64427	0.381736	0.190868	+	0.981616i	$0.438870\pi$
$402$	0	0
$403$	8.34365	0.415627
$404$	0	0
$405$	69.4461	3.45080
$406$	0	0
$407$	4.51943	0.224020
$408$	0	0
$409$	−2.12008	−0.104831	−0.0524157	−	0.998625i	$-0.516692\pi$
$409$	−2.12008	−0.104831	−0.0524157	+	0.998625i	$0.516692\pi$
$410$	0	0
$411$	30.1752	1.48843
$412$	0	0
$413$	6.50203	0.319944
$414$	0	0
$415$	42.2772	2.07531
$416$	0	0
$417$	59.2215	2.90009
$418$	0	0
$419$	16.3561	0.799049	0.399525	−	0.916722i	$-0.369175\pi$
$419$	16.3561	0.799049	0.399525	+	0.916722i	$0.369175\pi$
$420$	0	0
$421$	9.23960	0.450310	0.225155	−	0.974323i	$-0.427711\pi$
$421$	9.23960	0.450310	0.225155	+	0.974323i	$0.427711\pi$
$422$	0	0
$423$	9.15741	0.445248
$424$	0	0
$425$	15.4241	0.748180
$426$	0	0
$427$	−1.95309	−0.0945169
$428$	0	0
$429$	−5.61820	−0.271249
$430$	0	0
$431$	29.4110	1.41668	0.708339	−	0.705872i	$-0.249445\pi$
$431$	29.4110	1.41668	0.708339	+	0.705872i	$0.249445\pi$
$432$	0	0
$433$	20.3634	0.978602	0.489301	−	0.872115i	$-0.337252\pi$
$433$	20.3634	0.978602	0.489301	+	0.872115i	$0.337252\pi$
$434$	0	0
$435$	−46.6125	−2.23490
$436$	0	0
$437$	−0.767559	−0.0367173
$438$	0	0
$439$	−19.8359	−0.946718	−0.473359	−	0.880870i	$-0.656959\pi$
$439$	−19.8359	−0.946718	−0.473359	+	0.880870i	$0.656959\pi$
$440$	0	0
$441$	−49.8475	−2.37369
$442$	0	0
$443$	−8.27690	−0.393247	−0.196624	−	0.980479i	$-0.562998\pi$
$443$	−8.27690	−0.393247	−0.196624	+	0.980479i	$0.562998\pi$
$444$	0	0
$445$	−23.3414	−1.10649
$446$	0	0
$447$	11.4216	0.540223
$448$	0	0
$449$	34.5329	1.62971	0.814854	−	0.579666i	$-0.196817\pi$
$449$	34.5329	1.62971	0.814854	+	0.579666i	$0.196817\pi$
$450$	0	0
$451$	−12.2470	−0.576687
$452$	0	0
$453$	−7.49594	−0.352190
$454$	0	0
$455$	−2.24525	−0.105259
$456$	0	0
$457$	8.26068	0.386418	0.193209	−	0.981158i	$-0.438110\pi$
$457$	8.26068	0.386418	0.193209	+	0.981158i	$0.438110\pi$
$458$	0	0
$459$	−61.5539	−2.87309
$460$	0	0
$461$	22.0892	1.02880	0.514399	−	0.857551i	$-0.328015\pi$
$461$	22.0892	1.02880	0.514399	+	0.857551i	$0.328015\pi$
$462$	0	0
$463$	25.8766	1.20259	0.601293	−	0.799029i	$-0.294653\pi$
$463$	25.8766	1.20259	0.601293	+	0.799029i	$0.294653\pi$
$464$	0	0
$465$	54.0283	2.50550
$466$	0	0
$467$	24.6592	1.14109	0.570545	−	0.821266i	$-0.306732\pi$
$467$	24.6592	1.14109	0.570545	+	0.821266i	$0.306732\pi$
$468$	0	0
$469$	1.51084	0.0697643
$470$	0	0
$471$	7.59017	0.349736
$472$	0	0
$473$	−4.77075	−0.219359
$474$	0	0
$475$	9.52442	0.437010
$476$	0	0
$477$	89.2337	4.08573
$478$	0	0
$479$	12.5245	0.572260	0.286130	−	0.958191i	$-0.407631\pi$
$479$	12.5245	0.572260	0.286130	+	0.958191i	$0.407631\pi$
$480$	0	0
$481$	5.52474	0.251906
$482$	0	0
$483$	0.488012	0.0222053
$484$	0	0
$485$	−30.7817	−1.39772
$486$	0	0
$487$	−4.53327	−0.205422	−0.102711	−	0.994711i	$-0.532752\pi$
$487$	−4.53327	−0.205422	−0.102711	+	0.994711i	$0.532752\pi$
$488$	0	0
$489$	−20.7104	−0.936558
$490$	0	0
$491$	31.9573	1.44221	0.721106	−	0.692825i	$-0.243634\pi$
$491$	31.9573	1.44221	0.721106	+	0.692825i	$0.243634\pi$
$492$	0	0
$493$	21.3237	0.960369
$494$	0	0
$495$	−25.8997	−1.16410
$496$	0	0
$497$	−2.31739	−0.103949
$498$	0	0
$499$	14.2081	0.636043	0.318021	−	0.948084i	$-0.396982\pi$
$499$	14.2081	0.636043	0.318021	+	0.948084i	$0.396982\pi$
$500$	0	0
$501$	−72.9201	−3.25783
$502$	0	0
$503$	−26.5276	−1.18281	−0.591404	−	0.806375i	$-0.701426\pi$
$503$	−26.5276	−1.18281	−0.591404	+	0.806375i	$0.701426\pi$
$504$	0	0
$505$	−25.0622	−1.11525
$506$	0	0
$507$	35.0837	1.55812
$508$	0	0
$509$	−0.689637	−0.0305676	−0.0152838	−	0.999883i	$-0.504865\pi$
$509$	−0.689637	−0.0305676	−0.0152838	+	0.999883i	$0.504865\pi$
$510$	0	0
$511$	0.623952	0.0276020
$512$	0	0
$513$	−38.0096	−1.67817
$514$	0	0
$515$	29.3213	1.29205
$516$	0	0
$517$	−1.47405	−0.0648285
$518$	0	0
$519$	−33.6835	−1.47854
$520$	0	0
$521$	−3.32412	−0.145632	−0.0728161	−	0.997345i	$-0.523199\pi$
$521$	−3.32412	−0.145632	−0.0728161	+	0.997345i	$0.523199\pi$
$522$	0	0
$523$	10.3850	0.454104	0.227052	−	0.973883i	$-0.427091\pi$
$523$	10.3850	0.454104	0.227052	+	0.973883i	$0.427091\pi$
$524$	0	0
$525$	−6.05561	−0.264288
$526$	0	0
$527$	−24.7161	−1.07665
$528$	0	0
$529$	−22.9173	−0.996403
$530$	0	0
$531$	91.6868	3.97887
$532$	0	0
$533$	−14.9712	−0.648475
$534$	0	0
$535$	39.3721	1.70220
$536$	0	0
$537$	−24.1798	−1.04343
$538$	0	0
$539$	8.02384	0.345611
$540$	0	0
$541$	20.9154	0.899224	0.449612	−	0.893224i	$-0.351562\pi$
$541$	20.9154	0.899224	0.449612	+	0.893224i	$0.351562\pi$
$542$	0	0
$543$	−60.6329	−2.60201
$544$	0	0
$545$	53.7199	2.30111
$546$	0	0
$547$	−5.89179	−0.251915	−0.125957	−	0.992036i	$-0.540200\pi$
$547$	−5.89179	−0.251915	−0.125957	+	0.992036i	$0.540200\pi$
$548$	0	0
$549$	−27.5411	−1.17542
$550$	0	0
$551$	13.1674	0.560950
$552$	0	0
$553$	−4.63818	−0.197236
$554$	0	0
$555$	35.7747	1.51855
$556$	0	0
$557$	−31.9487	−1.35371	−0.676855	−	0.736116i	$-0.736658\pi$
$557$	−31.9487	−1.35371	−0.676855	+	0.736116i	$0.736658\pi$
$558$	0	0
$559$	−5.83196	−0.246665
$560$	0	0
$561$	16.6426	0.702651
$562$	0	0
$563$	9.73827	0.410419	0.205210	−	0.978718i	$-0.434212\pi$
$563$	9.73827	0.410419	0.205210	+	0.978718i	$0.434212\pi$
$564$	0	0
$565$	−13.0662	−0.549699
$566$	0	0
$567$	12.4728	0.523808
$568$	0	0
$569$	26.7550	1.12163	0.560813	−	0.827942i	$-0.310489\pi$
$569$	26.7550	1.12163	0.560813	+	0.827942i	$0.310489\pi$
$570$	0	0
$571$	13.4914	0.564599	0.282299	−	0.959326i	$-0.408903\pi$
$571$	13.4914	0.564599	0.282299	+	0.959326i	$0.408903\pi$
$572$	0	0
$573$	32.7836	1.36955
$574$	0	0
$575$	−1.02661	−0.0428128
$576$	0	0
$577$	−9.05659	−0.377031	−0.188515	−	0.982070i	$-0.560368\pi$
$577$	−9.05659	−0.377031	−0.188515	+	0.982070i	$0.560368\pi$
$578$	0	0
$579$	−27.7225	−1.15211
$580$	0	0
$581$	7.59316	0.315017
$582$	0	0
$583$	−14.3637	−0.594886
$584$	0	0
$585$	−31.6608	−1.30901
$586$	0	0
$587$	2.57333	0.106212	0.0531062	−	0.998589i	$-0.483088\pi$
$587$	2.57333	0.106212	0.0531062	+	0.998589i	$0.483088\pi$
$588$	0	0
$589$	−15.2622	−0.628870
$590$	0	0
$591$	−57.9647	−2.38435
$592$	0	0
$593$	7.58695	0.311559	0.155779	−	0.987792i	$-0.450211\pi$
$593$	7.58695	0.311559	0.155779	+	0.987792i	$0.450211\pi$
$594$	0	0
$595$	6.65102	0.272665
$596$	0	0
$597$	13.7072	0.560998
$598$	0	0
$599$	−24.5537	−1.00324	−0.501619	−	0.865089i	$-0.667262\pi$
$599$	−24.5537	−1.00324	−0.501619	+	0.865089i	$0.667262\pi$
$600$	0	0
$601$	38.8086	1.58304	0.791518	−	0.611145i	$-0.209291\pi$
$601$	38.8086	1.58304	0.791518	+	0.611145i	$0.209291\pi$
$602$	0	0
$603$	21.3048	0.867598
$604$	0	0
$605$	−28.0314	−1.13964
$606$	0	0
$607$	15.1125	0.613395	0.306698	−	0.951807i	$-0.400776\pi$
$607$	15.1125	0.613395	0.306698	+	0.951807i	$0.400776\pi$
$608$	0	0
$609$	−8.37180	−0.339242
$610$	0	0
$611$	−1.80193	−0.0728985
$612$	0	0
$613$	−45.8408	−1.85149	−0.925746	−	0.378147i	$-0.876562\pi$
$613$	−45.8408	−1.85149	−0.925746	+	0.378147i	$0.876562\pi$
$614$	0	0
$615$	−96.9441	−3.90916
$616$	0	0
$617$	−27.0879	−1.09052	−0.545259	−	0.838268i	$-0.683569\pi$
$617$	−27.0879	−1.09052	−0.545259	+	0.838268i	$0.683569\pi$
$618$	0	0
$619$	16.4472	0.661069	0.330535	−	0.943794i	$-0.392771\pi$
$619$	16.4472	0.661069	0.330535	+	0.943794i	$0.392771\pi$
$620$	0	0
$621$	4.09696	0.164405
$622$	0	0
$623$	−4.19220	−0.167957
$624$	0	0
$625$	−30.1069	−1.20428
$626$	0	0
$627$	10.2768	0.410417
$628$	0	0
$629$	−16.3657	−0.652544
$630$	0	0
$631$	−21.0499	−0.837984	−0.418992	−	0.907990i	$-0.637617\pi$
$631$	−21.0499	−0.837984	−0.418992	+	0.907990i	$0.637617\pi$
$632$	0	0
$633$	48.5340	1.92906
$634$	0	0
$635$	−31.9214	−1.26676
$636$	0	0
$637$	9.80867	0.388634
$638$	0	0
$639$	−32.6781	−1.29273
$640$	0	0
$641$	−45.6273	−1.80217	−0.901086	−	0.433641i	$-0.857229\pi$
$641$	−45.6273	−1.80217	−0.901086	+	0.433641i	$0.857229\pi$
$642$	0	0
$643$	−8.74422	−0.344838	−0.172419	−	0.985024i	$-0.555158\pi$
$643$	−8.74422	−0.344838	−0.172419	+	0.985024i	$0.555158\pi$
$644$	0	0
$645$	−37.7641	−1.48696
$646$	0	0
$647$	19.6773	0.773596	0.386798	−	0.922165i	$-0.373581\pi$
$647$	19.6773	0.773596	0.386798	+	0.922165i	$0.373581\pi$
$648$	0	0
$649$	−14.7586	−0.579326
$650$	0	0
$651$	9.70370	0.380318
$652$	0	0
$653$	38.8950	1.52208	0.761039	−	0.648706i	$-0.224689\pi$
$653$	38.8950	1.52208	0.761039	+	0.648706i	$0.224689\pi$
$654$	0	0
$655$	−37.2642	−1.45603
$656$	0	0
$657$	8.79851	0.343263
$658$	0	0
$659$	−22.0282	−0.858095	−0.429048	−	0.903282i	$-0.641151\pi$
$659$	−22.0282	−0.858095	−0.429048	+	0.903282i	$0.641151\pi$
$660$	0	0
$661$	−37.6441	−1.46419	−0.732093	−	0.681205i	$-0.761456\pi$
$661$	−37.6441	−1.46419	−0.732093	+	0.681205i	$0.761456\pi$
$662$	0	0
$663$	20.3446	0.790118
$664$	0	0
$665$	4.10702	0.159263
$666$	0	0
$667$	−1.41928	−0.0549548
$668$	0	0
$669$	32.5207	1.25732
$670$	0	0
$671$	4.43323	0.171143
$672$	0	0
$673$	42.9416	1.65528	0.827638	−	0.561262i	$-0.189684\pi$
$673$	42.9416	1.65528	0.827638	+	0.561262i	$0.189684\pi$
$674$	0	0
$675$	−50.8380	−1.95676
$676$	0	0
$677$	37.4409	1.43897	0.719486	−	0.694507i	$-0.244377\pi$
$677$	37.4409	1.43897	0.719486	+	0.694507i	$0.244377\pi$
$678$	0	0
$679$	−5.52852	−0.212165
$680$	0	0
$681$	38.9275	1.49171
$682$	0	0
$683$	1.08390	0.0414743	0.0207372	−	0.999785i	$-0.493399\pi$
$683$	1.08390	0.0414743	0.0207372	+	0.999785i	$0.493399\pi$
$684$	0	0
$685$	−27.3725	−1.04585
$686$	0	0
$687$	30.5701	1.16632
$688$	0	0
$689$	−17.5588	−0.668938
$690$	0	0
$691$	22.2008	0.844559	0.422279	−	0.906466i	$-0.361230\pi$
$691$	22.2008	0.844559	0.422279	+	0.906466i	$0.361230\pi$
$692$	0	0
$693$	−4.65169	−0.176703
$694$	0	0
$695$	−53.7209	−2.03775
$696$	0	0
$697$	44.3486	1.67982
$698$	0	0
$699$	−6.44608	−0.243813
$700$	0	0
$701$	1.43258	0.0541078	0.0270539	−	0.999634i	$-0.491387\pi$
$701$	1.43258	0.0541078	0.0270539	+	0.999634i	$0.491387\pi$
$702$	0	0
$703$	−10.1059	−0.381150
$704$	0	0
$705$	−11.6682	−0.439450
$706$	0	0
$707$	−4.50127	−0.169288
$708$	0	0
$709$	−35.2484	−1.32378	−0.661891	−	0.749600i	$-0.730246\pi$
$709$	−35.2484	−1.32378	−0.661891	+	0.749600i	$0.730246\pi$
$710$	0	0
$711$	−65.4042	−2.45285
$712$	0	0
$713$	1.64508	0.0616087
$714$	0	0
$715$	5.09637	0.190593
$716$	0	0
$717$	−46.0761	−1.72074
$718$	0	0
$719$	29.9828	1.11817	0.559086	−	0.829110i	$-0.311152\pi$
$719$	29.9828	1.11817	0.559086	+	0.829110i	$0.311152\pi$
$720$	0	0
$721$	5.26622	0.196124
$722$	0	0
$723$	−40.2909	−1.49844
$724$	0	0
$725$	17.6114	0.654073
$726$	0	0
$727$	−18.2631	−0.677342	−0.338671	−	0.940905i	$-0.609977\pi$
$727$	−18.2631	−0.677342	−0.338671	+	0.940905i	$0.609977\pi$
$728$	0	0
$729$	37.9871	1.40693
$730$	0	0
$731$	17.2758	0.638968
$732$	0	0
$733$	−32.3017	−1.19309	−0.596545	−	0.802579i	$-0.703460\pi$
$733$	−32.3017	−1.19309	−0.596545	+	0.802579i	$0.703460\pi$
$734$	0	0
$735$	63.5148	2.34278
$736$	0	0
$737$	−3.42938	−0.126323
$738$	0	0
$739$	28.5787	1.05128	0.525642	−	0.850706i	$-0.323825\pi$
$739$	28.5787	1.05128	0.525642	+	0.850706i	$0.323825\pi$
$740$	0	0
$741$	12.5628	0.461506
$742$	0	0
$743$	30.2706	1.11052	0.555260	−	0.831677i	$-0.312619\pi$
$743$	30.2706	1.11052	0.555260	+	0.831677i	$0.312619\pi$
$744$	0	0
$745$	−10.3607	−0.379588
$746$	0	0
$747$	107.073	3.91760
$748$	0	0
$749$	7.07138	0.258383
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	65.0708	2.37131
$754$	0	0
$755$	6.79970	0.247466
$756$	0	0
$757$	17.5444	0.637660	0.318830	−	0.947812i	$-0.396710\pi$
$757$	17.5444	0.637660	0.318830	+	0.947812i	$0.396710\pi$
$758$	0	0
$759$	−1.10771	−0.0402075
$760$	0	0
$761$	−5.01959	−0.181960	−0.0909800	−	0.995853i	$-0.529000\pi$
$761$	−5.01959	−0.181960	−0.0909800	+	0.995853i	$0.529000\pi$
$762$	0	0
$763$	9.64830	0.349292
$764$	0	0
$765$	93.7877	3.39090
$766$	0	0
$767$	−18.0415	−0.651442
$768$	0	0
$769$	−30.0687	−1.08431	−0.542153	−	0.840280i	$-0.682391\pi$
$769$	−30.0687	−1.08431	−0.542153	+	0.840280i	$0.682391\pi$
$770$	0	0
$771$	22.4756	0.809439
$772$	0	0
$773$	19.7999	0.712154	0.356077	−	0.934457i	$-0.384114\pi$
$773$	19.7999	0.712154	0.356077	+	0.934457i	$0.384114\pi$
$774$	0	0
$775$	−20.4133	−0.733268
$776$	0	0
$777$	6.42529	0.230506
$778$	0	0
$779$	27.3854	0.981183
$780$	0	0
$781$	5.26013	0.188222
$782$	0	0
$783$	−70.2829	−2.51171
$784$	0	0
$785$	−6.88518	−0.245743
$786$	0	0
$787$	−29.1549	−1.03926	−0.519630	−	0.854391i	$-0.673930\pi$
$787$	−29.1549	−1.03926	−0.519630	+	0.854391i	$0.673930\pi$
$788$	0	0
$789$	−53.0116	−1.88726
$790$	0	0
$791$	−2.34674	−0.0834405
$792$	0	0
$793$	5.41936	0.192447
$794$	0	0
$795$	−113.700	−4.03252
$796$	0	0
$797$	2.81614	0.0997529	0.0498765	−	0.998755i	$-0.484117\pi$
$797$	2.81614	0.0997529	0.0498765	+	0.998755i	$0.484117\pi$
$798$	0	0
$799$	5.33781	0.188838
$800$	0	0
$801$	−59.1153	−2.08874
$802$	0	0
$803$	−1.41628	−0.0499793
$804$	0	0
$805$	−0.442685	−0.0156026
$806$	0	0
$807$	−36.0078	−1.26754
$808$	0	0
$809$	−40.6062	−1.42764	−0.713819	−	0.700331i	$-0.753036\pi$
$809$	−40.6062	−1.42764	−0.713819	+	0.700331i	$0.753036\pi$
$810$	0	0
$811$	−25.4202	−0.892623	−0.446312	−	0.894878i	$-0.647263\pi$
$811$	−25.4202	−0.892623	−0.446312	+	0.894878i	$0.647263\pi$
$812$	0	0
$813$	−87.2343	−3.05944
$814$	0	0
$815$	18.7868	0.658073
$816$	0	0
$817$	10.6678	0.373220
$818$	0	0
$819$	−5.68641	−0.198699
$820$	0	0
$821$	25.6224	0.894227	0.447113	−	0.894477i	$-0.352452\pi$
$821$	25.6224	0.894227	0.447113	+	0.894477i	$0.352452\pi$
$822$	0	0
$823$	−7.71419	−0.268900	−0.134450	−	0.990920i	$-0.542927\pi$
$823$	−7.71419	−0.268900	−0.134450	+	0.990920i	$0.542927\pi$
$824$	0	0
$825$	13.7453	0.478550
$826$	0	0
$827$	−13.4859	−0.468951	−0.234475	−	0.972122i	$-0.575337\pi$
$827$	−13.4859	−0.468951	−0.234475	+	0.972122i	$0.575337\pi$
$828$	0	0
$829$	−8.19326	−0.284564	−0.142282	−	0.989826i	$-0.545444\pi$
$829$	−8.19326	−0.284564	−0.142282	+	0.989826i	$0.545444\pi$
$830$	0	0
$831$	−33.0304	−1.14581
$832$	0	0
$833$	−29.0559	−1.00673
$834$	0	0
$835$	66.1472	2.28912
$836$	0	0
$837$	81.4645	2.81582
$838$	0	0
$839$	−29.2073	−1.00835	−0.504175	−	0.863602i	$-0.668203\pi$
$839$	−29.2073	−1.00835	−0.504175	+	0.863602i	$0.668203\pi$
$840$	0	0
$841$	−4.65240	−0.160428
$842$	0	0
$843$	−21.7105	−0.747750
$844$	0	0
$845$	−31.8251	−1.09482
$846$	0	0
$847$	−5.03456	−0.172989
$848$	0	0
$849$	−19.6401	−0.674047
$850$	0	0
$851$	1.08929	0.0373402
$852$	0	0
$853$	−45.3761	−1.55365	−0.776825	−	0.629717i	$-0.783171\pi$
$853$	−45.3761	−1.55365	−0.776825	+	0.629717i	$0.783171\pi$
$854$	0	0
$855$	57.9141	1.98062
$856$	0	0
$857$	−1.17185	−0.0400296	−0.0200148	−	0.999800i	$-0.506371\pi$
$857$	−1.17185	−0.0400296	−0.0200148	+	0.999800i	$0.506371\pi$
$858$	0	0
$859$	43.4863	1.48373	0.741867	−	0.670547i	$-0.233940\pi$
$859$	43.4863	1.48373	0.741867	+	0.670547i	$0.233940\pi$
$860$	0	0
$861$	−17.4116	−0.593384
$862$	0	0
$863$	−14.6408	−0.498378	−0.249189	−	0.968455i	$-0.580164\pi$
$863$	−14.6408	−0.498378	−0.249189	+	0.968455i	$0.580164\pi$
$864$	0	0
$865$	30.5549	1.03890
$866$	0	0
$867$	−5.40620	−0.183604
$868$	0	0
$869$	10.5280	0.357137
$870$	0	0
$871$	−4.19222	−0.142048
$872$	0	0
$873$	−77.9590	−2.63851
$874$	0	0
$875$	−2.20213	−0.0744457
$876$	0	0
$877$	21.7239	0.733562	0.366781	−	0.930307i	$-0.380460\pi$
$877$	21.7239	0.733562	0.366781	+	0.930307i	$0.380460\pi$
$878$	0	0
$879$	43.2790	1.45976
$880$	0	0
$881$	−50.2337	−1.69242	−0.846209	−	0.532852i	$-0.821120\pi$
$881$	−50.2337	−1.69242	−0.846209	+	0.532852i	$0.821120\pi$
$882$	0	0
$883$	25.5432	0.859598	0.429799	−	0.902925i	$-0.358584\pi$
$883$	25.5432	0.859598	0.429799	+	0.902925i	$0.358584\pi$
$884$	0	0
$885$	−116.826	−3.92705
$886$	0	0
$887$	41.6041	1.39693	0.698465	−	0.715644i	$-0.253867\pi$
$887$	41.6041	1.39693	0.698465	+	0.715644i	$0.253867\pi$
$888$	0	0
$889$	−5.73322	−0.192286
$890$	0	0
$891$	−28.3113	−0.948465
$892$	0	0
$893$	3.29611	0.110300
$894$	0	0
$895$	21.9339	0.733170
$896$	0	0
$897$	−1.35411	−0.0452126
$898$	0	0
$899$	−28.2211	−0.941228
$900$	0	0
$901$	52.0139	1.73283
$902$	0	0
$903$	−6.78259	−0.225710
$904$	0	0
$905$	55.0012	1.82830
$906$	0	0
$907$	50.3438	1.67164	0.835819	−	0.549006i	$-0.184993\pi$
$907$	50.3438	1.67164	0.835819	+	0.549006i	$0.184993\pi$
$908$	0	0
$909$	−63.4735	−2.10528
$910$	0	0
$911$	52.2663	1.73166	0.865830	−	0.500338i	$-0.166791\pi$
$911$	52.2663	1.73166	0.865830	+	0.500338i	$0.166791\pi$
$912$	0	0
$913$	−17.2353	−0.570405
$914$	0	0
$915$	35.0924	1.16012
$916$	0	0
$917$	−6.69280	−0.221016
$918$	0	0
$919$	5.31746	0.175407	0.0877034	−	0.996147i	$-0.472047\pi$
$919$	5.31746	0.175407	0.0877034	+	0.996147i	$0.472047\pi$
$920$	0	0
$921$	−95.8290	−3.15767
$922$	0	0
$923$	6.43019	0.211652
$924$	0	0
$925$	−13.5166	−0.444424
$926$	0	0
$927$	74.2603	2.43903
$928$	0	0
$929$	−54.8514	−1.79961	−0.899807	−	0.436287i	$-0.856293\pi$
$929$	−54.8514	−1.79961	−0.899807	+	0.436287i	$0.856293\pi$
$930$	0	0
$931$	−17.9421	−0.588027
$932$	0	0
$933$	9.21513	0.301690
$934$	0	0
$935$	−15.0968	−0.493718
$936$	0	0
$937$	36.6600	1.19763	0.598815	−	0.800887i	$-0.295638\pi$
$937$	36.6600	1.19763	0.598815	+	0.800887i	$0.295638\pi$
$938$	0	0
$939$	−20.1368	−0.657141
$940$	0	0
$941$	34.1635	1.11370	0.556849	−	0.830614i	$-0.312010\pi$
$941$	34.1635	1.11370	0.556849	+	0.830614i	$0.312010\pi$
$942$	0	0
$943$	−2.95180	−0.0961239
$944$	0	0
$945$	−21.9218	−0.713117
$946$	0	0
$947$	13.9428	0.453081	0.226541	−	0.974002i	$-0.427258\pi$
$947$	13.9428	0.453081	0.226541	+	0.974002i	$0.427258\pi$
$948$	0	0
$949$	−1.73131	−0.0562008
$950$	0	0
$951$	−8.14628	−0.264161
$952$	0	0
$953$	−56.1061	−1.81745	−0.908727	−	0.417391i	$-0.862945\pi$
$953$	−56.1061	−1.81745	−0.908727	+	0.417391i	$0.862945\pi$
$954$	0	0
$955$	−29.7386	−0.962318
$956$	0	0
$957$	19.0027	0.614270
$958$	0	0
$959$	−4.91621	−0.158753
$960$	0	0
$961$	1.71095	0.0551919
$962$	0	0
$963$	99.7153	3.21328
$964$	0	0
$965$	25.1476	0.809530
$966$	0	0
$967$	−48.0536	−1.54530	−0.772651	−	0.634832i	$-0.781070\pi$
$967$	−48.0536	−1.54530	−0.772651	+	0.634832i	$0.781070\pi$
$968$	0	0
$969$	−37.2144	−1.19550
$970$	0	0
$971$	11.9474	0.383409	0.191705	−	0.981453i	$-0.438598\pi$
$971$	11.9474	0.383409	0.191705	+	0.981453i	$0.438598\pi$
$972$	0	0
$973$	−9.64849	−0.309316
$974$	0	0
$975$	16.8028	0.538121
$976$	0	0
$977$	−36.7336	−1.17521	−0.587606	−	0.809147i	$-0.699930\pi$
$977$	−36.7336	−1.17521	−0.587606	+	0.809147i	$0.699930\pi$
$978$	0	0
$979$	9.51566	0.304122
$980$	0	0
$981$	136.053	4.34384
$982$	0	0
$983$	6.39913	0.204100	0.102050	−	0.994779i	$-0.467460\pi$
$983$	6.39913	0.204100	0.102050	+	0.994779i	$0.467460\pi$
$984$	0	0
$985$	52.5808	1.67536
$986$	0	0
$987$	−2.09566	−0.0667055
$988$	0	0
$989$	−1.14986	−0.0365634
$990$	0	0
$991$	8.26235	0.262462	0.131231	−	0.991352i	$-0.458107\pi$
$991$	8.26235	0.262462	0.131231	+	0.991352i	$0.458107\pi$
$992$	0	0
$993$	34.6711	1.10025
$994$	0	0
$995$	−12.4340	−0.394185
$996$	0	0
$997$	4.66972	0.147892	0.0739458	−	0.997262i	$-0.476441\pi$
$997$	4.66972	0.147892	0.0739458	+	0.997262i	$0.476441\pi$
$998$	0	0
$999$	53.9416	1.70664

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.1	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.1	✓	50	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-3.22705 q^{3} +2.92731 q^{5} +0.525757 q^{7} +7.41384 q^{9} +O(q^{10})\)	\(q-3.22705 q^{3} +2.92731 q^{5} +0.525757 q^{7} +7.41384 q^{9} -1.19339 q^{11} -1.45885 q^{13} -9.44658 q^{15} +4.32149 q^{17} +2.66853 q^{19} -1.69664 q^{21} -0.287634 q^{23} +3.56917 q^{25} -14.2437 q^{27} +4.93433 q^{29} -5.71935 q^{31} +3.85112 q^{33} +1.53906 q^{35} -3.78706 q^{37} +4.70777 q^{39} +10.2624 q^{41} +3.99765 q^{43} +21.7026 q^{45} +1.23518 q^{47} -6.72358 q^{49} -13.9457 q^{51} +12.0361 q^{53} -3.49342 q^{55} -8.61147 q^{57} +12.3670 q^{59} -3.71482 q^{61} +3.89788 q^{63} -4.27050 q^{65} +2.87365 q^{67} +0.928209 q^{69} -4.40772 q^{71} +1.18677 q^{73} -11.5179 q^{75} -0.627433 q^{77} -8.82191 q^{79} +23.7235 q^{81} +14.4423 q^{83} +12.6504 q^{85} -15.9233 q^{87} -7.97365 q^{89} -0.767000 q^{91} +18.4566 q^{93} +7.81162 q^{95} -10.5153 q^{97} -8.84759 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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